General solution of a PDE

Question

I am trying to solve for x>0 and y>0 the following PDE:

$$ x^2 u_{xx} -y^2 u_{yy} +x u_{x} - y u_{y} =0 $$

The characteristics are $$ \frac{dy}{dx} = \pm \frac{y}{x} $$ so I get

$\xi = \frac{y}{x} $ and $\eta = \frac{-y}{x} $

After the change of variables the PDE is getting the form

$$ \frac{-2y}{x} u_{\xi} + \frac{2y}{x} u_{\eta} = 0 $$

After that I can get

$$-2(\xi u_{\xi} +\eta u_{\eta})=0 $$ or $$ 2\eta (u_{\xi} - u_{\eta}) =0 $$ and take different solutions.

Where am I getting it wrong?

Thanks

Answer 1

The mistake is in $\eta=-\frac{y}{x}$ which should be $\eta=xy$.

With the change of variables $$\begin{cases} \xi=\frac{y}{x}\\ \eta=xy \end{cases}$$ you will get the general solution : $$u(x,y)=F\left(\frac{y}{x}\right)+G\big(xy\big)$$ with arbitrary functions $F$ and $G$.